The 67-year-old is scheduled to hold talks with club chiefs early next week after so far failing to confirm whether he intends to stay on for another couple of seasons.
Wenger admitted prior to Saturday's 2-1 win at Wembley Stadium that he was upset by the "disgraceful" criticism he has faced during his most difficult campaign to date, but Coquelin hopes to see his compatriot stick around for at least another season.
"I think it's a pleasure for everyone to offer this trophy to the coach," he told reporters. "We will know in the next few days what he will do. Regardless of his decision, I think it was important for him to win this FA Cup. We are all proud to have done it.
"We hope he will stay, but it's not our decision, he is the only one who can take it. We'll see in the next few days. But we are happy to have won. I said it many times, when we hear the word 'Arsenal', the first name we think is Arsene Wenger.
"He did a lot of things for the club. I think fans should not forget it. Even though it has been a bit difficult in recent seasons. The fans want more. But I also know that the coach wants more and he will do everything to win trophies and we hope he will be there next season."